Moment Varieties of Gaussian Mixtures

TL;DR

This paper explores the algebraic structure of moment varieties for Gaussian mixtures, providing new computational tools to recover mixture parameters from moments, extending classical algebraic geometry concepts.

Contribution

It introduces a detailed algebraic analysis of Gaussian mixture moment varieties and computes their defining ideals, bridging algebraic geometry with statistical parameter recovery.

Findings

Computed ideals for 5-dimensional moment varieties of two univariate Gaussians.

Extended algebraic geometric objects to Gaussian mixture models.

Compared algebraic methods with maximum likelihood approaches.

Abstract

The points of a moment variety are the vectors of all moments up to some order of a family of probability distributions. We study this variety for mixtures of Gaussians. Following up on Pearson's classical work from 1894, we apply current tools from computational algebra to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all covariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representing mixtures of two univariate Gaussians, and we offer a comparison to the maximum likelihood approach.